Question

a plane is flying within sight of the gateway arch in st. louis, missouri, at an elevation of 40,000 ft. the pilot would like to estimate her distance from the gateway arch. she finds that the angle of depression to a point on the ground below the arch is 19°. (round your answers to the nearest foot.) (a) what is the distance between the plane and the base of the arch? ft (b) what is the distance between a point on the ground directly below the plane and the base of the arch? ft resources read it

Explanation:

Step1: Identify the right - triangle relationships

We have a right - triangle where the height of the plane (opposite side) is $h = 40000$ ft and the angle of depression is $19^{\circ}$. The angle of depression is equal to the angle of elevation from the point on the ground to the plane.

Step2: Find the distance between the plane and the base of the arch (hypotenuse)

We use the sine function $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Let the distance between the plane and the base of the arch be $d_1$. So, $\sin(19^{\circ})=\frac{40000}{d_1}$. Then $d_1=\frac{40000}{\sin(19^{\circ})}$.
$d_1=\frac{40000}{0.3256}\approx122850$ ft

Step3: Find the distance between a point on the ground directly below the plane and the base of the arch (adjacent side)

We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the distance between a point on the ground directly below the plane and the base of the arch be $d_2$. So, $\tan(19^{\circ})=\frac{40000}{d_2}$. Then $d_2 = \frac{40000}{\tan(19^{\circ})}$.
$d_2=\frac{40000}{0.3443}\approx116178$ ft

Answer:

(a) 122850 ft
(b) 116178 ft